Astrodynamics (AERO0024) 10. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Motivation 2 6. Interplanetary Trajectories 6.1 Patched conic method 6.2 Lambert’s problem 6.3 Gravity assist 3 6. Interplanetary Trajectories 6.1 Patched conic method 6.1.1 Problem statement 6.1.2 Sphere of influence 6.1.3 Planetary departure 6.1.4 Hohmann transfer 6.1.5 Planetary arrival 6.1.6 Sensitivity analysis and launch windows 6.1.7 Examples 4 ? Hint #1: design the Earth-Mars transfer using known concepts Hint #2: division into simpler problems Hint #3: patched conic method 5 What Transfer Orbit ? Constraints ? Dep. “V1” Arr. Dep. Arr. Dep. “V2” Arr. Motion in the heliocentric reference frame6 Planetary Departure ? Constraints ? v v Earth/ Sun V ? 1 Motion in the planetary reference frame7 Planetary Arrival ? Similar Reasoning Transfer ellipse SOI SOI Arrival Departure hyperbola hyperbola 8 Patched Conic Method Three conics to patch: 1. Outbound hyperbola (departure) 2. The Hohmann transfer ellipse (interplanetary travel) 3. Inbound hyperbola (arrival) 6.1.1 Problem statement 9 Patched Conic Method Approximate method that analyzes a mission as a sequence of 2-body problems, with one body always being the spacecraft. If the spacecraft is close enough to one celestial body, the gravitational forces due to other planets can be neglected. The region inside of which the approximation is valid is called the sphere of influence (SOI) of the celestial body. If the spacecraft is not inside the SOI of a planet, it is considered to be in orbit around the sun. 6.1.1 Problem statement 10 Patched Conic Method Very useful for preliminary mission design (delta-v requirements and flight times). But actual mission design and execution employ the most accurate possible numerical integration techniques. 6.1.1 Problem statement 11 Sphere of Influence (SOI) ? Let’s assume that a spacecraft is within the Earth’s SOI if the gravitational force due to Earth is larger than the gravitational force due to the sun. GmE m sat Gm S m sat 22 rrE,, sat S sat 5 rE, sat 2.5 10 km 6.1.2 Sphere of influence 12 Sphere of Influence (SOI) Third body: sun or planet m Spacecraft d j m2 r ρ d ρ rr3 Gm j 3 3 rd m1 Central body: Disturbing O sun or planet function (L04) 6.1.2 Sphere of influence 13 If the Spacecraft Orbits the Planet G m m p:planet pv rsv rsp rr Gm v: vehicle pv3 pv s 3 3 rpv r sv r sp s:sun rpv A p P s Perturbation acceleration due Primary to the sun gravitational acceleration due to the planet 6.1.2 Sphere of influence 14 If the Spacecraft Orbits the Sun G msv m rrpv sp p:planet rr Gm v: vehicle sv3 sv p 3 3 rsv r pv r sp s:sun rsv A s P p Perturbation acceleration due Primary to the planet gravitational acceleration due to the sun 6.1.2 Sphere of influence 15 SOI: Correct Definition due to Laplace It is the surface along which the magnitudes of the acceleration satisfy: P P p s AAsp Measure of the planet’s Measure of the deviation of influence on the orbit of the the vehicle’s orbit from the vehicle relative to the sun Keplerian orbit arising from the planet acting by itself 2 5 mp rrSOI sp ms 6.1.2 Sphere of influence 16 SOI: Correct Definition due to Laplace P P If p s the spacecraft is inside the SOI of the planet. AAsp Ap The previous (incorrect) definition was 1 As The moon lied outside the SOI and was in orbit about the sun like an asteroid ! 6.1.2 Sphere of influence 17 SOI Radii SOI radius Planet SOI Radius (km) (body radii) Mercury 1.13x105 45 Venus 6.17x105 100 OK ! Earth 9.24x105 145 Mars 5.74x105 170 Jupiter 4.83x107 677 Neptune 8.67x107 3886 6.1.2 Sphere of influence 18 Validity of the Patched Conic Method The Earth’s SOI is 145 Earth radii. This is extremely large compared to the size of the Earth: The velocity relative to the planet on an escape hyperbola is considered to be the hyperbolic excess velocity vector. vvSOI This is extremely small with respect to 1AU: During the elliptic transfer, the spacecraft is considered to be under the influence of the Sun’s gravity only. In other words, it follows an unperturbed Keplerian orbit around the Sun. 6.1.2 Sphere of influence 19 Outbound Hyperbola The spacecraft necessarily escapes using a hyperbolic trajectory relative to the planet. When this velocity vector is added to the planet’s heliocentric velocity, the result is the spacecraft’s heliocentric velocity on the Hyperbolic excess interplanetary elliptic transfer orbit at the speed SOI in the solar system. 2 2 2 2 2 Lecture 02: v v vesc v SOI v esc Is vSOI the velocity on the transfer orbit ? 6.1.3 Planetary departure 20 Magnitude of VSOI The velocity vD of the spacecraft relative to the sun is imposed by the Hohmann transfer (i.e., velocity on the transfer orbit). H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 6.1.3 Planetary departure 21 Magnitude of VSOI By subtracting the known value of the velocity v1 of the planet relative to the sun, one obtains the hyperbolic excess speed on the Earth escape hyperbola. 2R v v v sun 2 1 v SOI D 1 RRR 1 1 2 Imposed Lecture05 Known 6.1.3 Planetary departure 22 Direction of VSOI What should be the direction of vSOI ? For a Hohmann transfer, it should be parallel to v1. 6.1.3 Planetary departure 23 Parking Orbit A spacecraft is ordinary launched into an interplanetary trajectory from a circular parking orbit. Its radius equals the periapse radius rp of the departure hyperbola. H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 6.1.3 Planetary departure 24 ΔV Magnitude and Location 2 2 h rvp Lecture 02: r e 1 p (1 e ) known known v 2 a e 1 2 2 h h rp v 2 r h 1 v p a e2 1 h 1 1 v vpc v cos rrpp e 6.1.3 Planetary departure 25 Planetary Departure: Graphically Departure to outer or inner planet ? H. Curtis, Orbital Mechanics for Engineering Students, Elsevier. 6.1.3 Planetary departure 26 Circular, Coplanar Orbits for Most Planets Inclination of the orbit Planet Eccentricity to the ecliptic plane Mercury 7.00º 0.206 Venus 3.39º 0.007 Earth 0.00º 0.017 Mars 1.85º 0.094 Jupiter 1.30º 0.049 Saturn 2.48º 0.056 Uranus 0.77º 0.046 Neptune 1.77º 0.011 Pluto 17.16º 0.244 6.1.4 Hohmann transfer 27 Governing Equations vD sun 2R2 vv 1 v1 D 1 RRR 1 1 2 sun 2R1 vA vv2 A 1 RRR v2 2 1 2 Signs ? v2 - vA, vD - v1 >0 for transfer to an outer planet v2 - vA, vD - v1 <0 for transfer to an inner planet 6.1.4 Hohmann transfer 28 Schematically Transfer to outer planet Transfer to inner planet 6.1.4 Hohmann transfer 29 Arrival at an Outer Planet For an outer planet, the spacecraft’s heliocentric approach velocity vA is smaller in magnitude than that of the planet v2. v2 v vA vv2 A v 2 and v have opposite signs. 6.1.5 Planetary arrival 30 Arrival at an Outer Planet The spacecraft crosses the forward portion of the SOI 6.1.5 Planetary arrival 31 Enter into an Elliptic Orbit If the intent is to go into orbit around the planet, then must be chosen so that the v burn at periapse will occur at the correct altitude above the planet. 2 rp 1 2 rvp h(1 e )2 2 (1 e ) v vp,, hyp v p capture v rp r p r p r p 6.1.5 Planetary arrival 32 Planetary Flyby 2 rvp Otherwise, the specacraft will simplye continue1 past periapse on a flyby trajectory exiting the SOI with the same relative speed v it entered but with the velocity vector rotated through the turn angle . 1 2sin1 e 6.1.5 Planetary arrival 33 Sensitivity Analysis: Departure The maneuver occurs well within the SOI, which is just a point on the scale of the solar system. One may therefore ask what effects small errors in position and velocity (rp and vp) at the maneuver point have on the trajectory (target radius R2 of the heliocentric Hohmann transfer ellipse). 21 v R 2 r r v 21p p p 2 R2 Rv1 D vD v r p r p v D v p 1 2sun 6.1.6 Sensitivity analysis and launch windows 34 Sensitivity Analysis: Earth-Mars, 300km Orbit 11 3 2 3 2 sun 1.327 10 km /ss ,1 398600 km / 66 R12149.6 10 km, R 227.9 10 km, rp 6678 km vD 32.73 km / s , v 2.943 km / s R rv 2 3.127pp 6.708 R2 rpp v A 0.01% variation in the burnout speed vp changes the target radius by 0.067% or 153000 km. A 0.01% variation in burnout radius rp (670 m !) produces an error over 70000 km. 6.1.6 Sensitivity analysis and launch windows 35 Sensitivity Analysis: Launch Errors Ariane V Trajectory correction maneuvers are clearly mandatory. 6.1.6 Sensitivity analysis and launch windows 36 Sensitivity Analysis: Arrival The heliocentric velocity of Mars in its orbit is roughly 24km/s.